Theorem llynlly 22085

Description: A locally 𝐴 space is n-locally 𝐴: the "n-locally" predicate is the weaker notion. (Contributed by Mario Carneiro, 2-Mar-2015.)

Assertion

Ref	Expression
llynlly	⊢ (𝐽 ∈ Locally 𝐴 → 𝐽 ∈ 𝑛-Locally 𝐴)

Proof of Theorem llynlly

Dummy variables 𝑢 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		llytop 22080	. 2 ⊢ (𝐽 ∈ Locally 𝐴 → 𝐽 ∈ Top)
2		llyi 22082	. . . . 5 ⊢ ((𝐽 ∈ Locally 𝐴 ∧ 𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥) → ∃𝑢 ∈ 𝐽 (𝑢 ⊆ 𝑥 ∧ 𝑦 ∈ 𝑢 ∧ (𝐽 ↾t 𝑢) ∈ 𝐴))
3		simpl1 1187	. . . . . . . 8 ⊢ (((𝐽 ∈ Locally 𝐴 ∧ 𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥) ∧ (𝑢 ∈ 𝐽 ∧ (𝑢 ⊆ 𝑥 ∧ 𝑦 ∈ 𝑢 ∧ (𝐽 ↾t 𝑢) ∈ 𝐴))) → 𝐽 ∈ Locally 𝐴)
4	3, 1	syl 17	. . . . . . 7 ⊢ (((𝐽 ∈ Locally 𝐴 ∧ 𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥) ∧ (𝑢 ∈ 𝐽 ∧ (𝑢 ⊆ 𝑥 ∧ 𝑦 ∈ 𝑢 ∧ (𝐽 ↾t 𝑢) ∈ 𝐴))) → 𝐽 ∈ Top)
5		simprl 769	. . . . . . 7 ⊢ (((𝐽 ∈ Locally 𝐴 ∧ 𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥) ∧ (𝑢 ∈ 𝐽 ∧ (𝑢 ⊆ 𝑥 ∧ 𝑦 ∈ 𝑢 ∧ (𝐽 ↾t 𝑢) ∈ 𝐴))) → 𝑢 ∈ 𝐽)
6		simprr2 1218	. . . . . . 7 ⊢ (((𝐽 ∈ Locally 𝐴 ∧ 𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥) ∧ (𝑢 ∈ 𝐽 ∧ (𝑢 ⊆ 𝑥 ∧ 𝑦 ∈ 𝑢 ∧ (𝐽 ↾t 𝑢) ∈ 𝐴))) → 𝑦 ∈ 𝑢)
7		opnneip 21727	. . . . . . 7 ⊢ ((𝐽 ∈ Top ∧ 𝑢 ∈ 𝐽 ∧ 𝑦 ∈ 𝑢) → 𝑢 ∈ ((nei‘𝐽)‘{𝑦}))
8	4, 5, 6, 7	syl3anc 1367	. . . . . 6 ⊢ (((𝐽 ∈ Locally 𝐴 ∧ 𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥) ∧ (𝑢 ∈ 𝐽 ∧ (𝑢 ⊆ 𝑥 ∧ 𝑦 ∈ 𝑢 ∧ (𝐽 ↾t 𝑢) ∈ 𝐴))) → 𝑢 ∈ ((nei‘𝐽)‘{𝑦}))
9		simprr1 1217	. . . . . . 7 ⊢ (((𝐽 ∈ Locally 𝐴 ∧ 𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥) ∧ (𝑢 ∈ 𝐽 ∧ (𝑢 ⊆ 𝑥 ∧ 𝑦 ∈ 𝑢 ∧ (𝐽 ↾t 𝑢) ∈ 𝐴))) → 𝑢 ⊆ 𝑥)
10		velpw 4544	. . . . . . 7 ⊢ (𝑢 ∈ 𝒫 𝑥 ↔ 𝑢 ⊆ 𝑥)
11	9, 10	sylibr 236	. . . . . 6 ⊢ (((𝐽 ∈ Locally 𝐴 ∧ 𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥) ∧ (𝑢 ∈ 𝐽 ∧ (𝑢 ⊆ 𝑥 ∧ 𝑦 ∈ 𝑢 ∧ (𝐽 ↾t 𝑢) ∈ 𝐴))) → 𝑢 ∈ 𝒫 𝑥)
12	8, 11	elind 4171	. . . . 5 ⊢ (((𝐽 ∈ Locally 𝐴 ∧ 𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥) ∧ (𝑢 ∈ 𝐽 ∧ (𝑢 ⊆ 𝑥 ∧ 𝑦 ∈ 𝑢 ∧ (𝐽 ↾t 𝑢) ∈ 𝐴))) → 𝑢 ∈ (((nei‘𝐽)‘{𝑦}) ∩ 𝒫 𝑥))
13		simprr3 1219	. . . . 5 ⊢ (((𝐽 ∈ Locally 𝐴 ∧ 𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥) ∧ (𝑢 ∈ 𝐽 ∧ (𝑢 ⊆ 𝑥 ∧ 𝑦 ∈ 𝑢 ∧ (𝐽 ↾t 𝑢) ∈ 𝐴))) → (𝐽 ↾t 𝑢) ∈ 𝐴)
14	2, 12, 13	reximssdv 3276	. . . 4 ⊢ ((𝐽 ∈ Locally 𝐴 ∧ 𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥) → ∃𝑢 ∈ (((nei‘𝐽)‘{𝑦}) ∩ 𝒫 𝑥)(𝐽 ↾t 𝑢) ∈ 𝐴)
15	14	3expb 1116	. . 3 ⊢ ((𝐽 ∈ Locally 𝐴 ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥)) → ∃𝑢 ∈ (((nei‘𝐽)‘{𝑦}) ∩ 𝒫 𝑥)(𝐽 ↾t 𝑢) ∈ 𝐴)
16	15	ralrimivva 3191	. 2 ⊢ (𝐽 ∈ Locally 𝐴 → ∀𝑥 ∈ 𝐽 ∀𝑦 ∈ 𝑥 ∃𝑢 ∈ (((nei‘𝐽)‘{𝑦}) ∩ 𝒫 𝑥)(𝐽 ↾t 𝑢) ∈ 𝐴)
17		isnlly 22077	. 2 ⊢ (𝐽 ∈ 𝑛-Locally 𝐴 ↔ (𝐽 ∈ Top ∧ ∀𝑥 ∈ 𝐽 ∀𝑦 ∈ 𝑥 ∃𝑢 ∈ (((nei‘𝐽)‘{𝑦}) ∩ 𝒫 𝑥)(𝐽 ↾t 𝑢) ∈ 𝐴))
18	1, 16, 17	sylanbrc 585	1 ⊢ (𝐽 ∈ Locally 𝐴 → 𝐽 ∈ 𝑛-Locally 𝐴)

Syntax hints:   → wi 4   ∧ wa 398   ∧ w3a 1083   ∈ wcel 2114  ∀wral 3138  ∃wrex 3139   ∩ cin 3935   ⊆ wss 3936  𝒫 cpw 4539  {csn 4567  ‘cfv 6355  (class class class)co 7156   ↾_t crest 16694  Topctop 21501  neicnei 21705  Locally clly 22072  𝑛-Locally cnlly 22073

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-rep 5190  ax-sep 5203  ax-nul 5210  ax-pow 5266  ax-pr 5330

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-reu 3145  df-rab 3147  df-v 3496  df-sbc 3773  df-csb 3884  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-pw 4541  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4839  df-iun 4921  df-br 5067  df-opab 5129  df-mpt 5147  df-id 5460  df-xp 5561  df-rel 5562  df-cnv 5563  df-co 5564  df-dm 5565  df-rn 5566  df-res 5567  df-ima 5568  df-iota 6314  df-fun 6357  df-fn 6358  df-f 6359  df-f1 6360  df-fo 6361  df-f1o 6362  df-fv 6363  df-ov 7159  df-top 21502  df-nei 21706  df-lly 22074  df-nlly 22075

This theorem is referenced by:  llyssnlly  22086  efmndtmd  22709